Operator-valued slice hyperholomorphic functions

In this chapter we introduce slice hyperholomorphic functions with values in a quaternionic Banach space. As in the complex case, there are two equivalent notions, namely weak and strong slice hyperholomorphicity. In order to properly define a multiplication between slice hyperholomorphic functions, we give a third characterization in terms of the Cauchy–Riemann system. Operator-valued functions can be obtained by using the so-called S-functional calculus. This calculus is associated with the notions of S-spectrum and S-resolvent, which are introduced and studied. We also present some hyperholomorphic extension results and, finally, we study the Hilbert-space-valued quaternionic Hardy space of the ball and its backward-shift invariant subspaces.